Simplifying Rational Expressions

What Is a Rational Expression?

A rational expression is a fraction with polynomials in the numerator and denominator:

$\frac{x^2 - 9}{x^2 + 5x + 6}$

Simplifying by Factoring and Cancelling

1. Factor the numerator.
2. Factor the denominator.
3. Cancel common factors.

Example: $\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x+3)(x-3)}{(x+3)(x+2)} = \frac{x-3}{x+2}$

(Valid for $x \neq -3$ and $x \neq -2$)

Restrictions on the Variable

The denominator cannot equal zero. After cancelling, note the original restrictions.

For $\frac{x^2 - 9}{x^2 + 5x + 6}$: $x \neq -3$ and $x \neq -2$.

Multiplying Rational Expressions

$\frac{A}{B} \times \frac{C}{D} = \frac{AC}{BD}$

Factor everything first, then cancel before multiplying:

$\frac{x+2}{x^2-1} \times \frac{x-1}{x+2} = \frac{(x+2)}{(x+1)(x-1)} \times \frac{(x-1)}{(x+2)} = \frac{1}{x+1}$

Dividing Rational Expressions

Flip the second fraction and multiply:

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$

Adding and Subtracting Rational Expressions

Find a common denominator, combine numerators:

$\frac{3}{x+1} + \frac{2}{x-1} = \frac{3(x-1) + 2(x+1)}{(x+1)(x-1)} = \frac{5x - 1}{x^2 - 1}$

Tip 1: Factor EVERYTHING First

Never cancel individual terms — only cancel common factors. $\frac{x^2 + x}{x} = \frac{x(x+1)}{x} = x + 1$, not $x + 1$ from cancelling the $x^2$.

Tip 2: Cancel Before Multiplying

Cross-cancelling between numerators and denominators keeps the numbers manageable.

Tip 3: Don't Cancel Terms — Cancel Factors

$\frac{x + 3}{x + 5} \neq \frac{3}{5}$. You can only cancel when the entire numerator and denominator share a common FACTOR.

Tip 4: State Restrictions

The SAT may ask for values where an expression is undefined. Set each denominator factor equal to zero.

Tip 5: LCD for Addition/Subtraction

Finding the least common denominator (LCD) is key for adding/subtracting rational expressions.